There is no law or directive that says that the forwarder in this instance must complete the export entry but if you have contracted with a freight forwarder to arrange the export then they are really only asking for you to supply the documentation/information needed in order to submit the export declaration. If you really wanted to you could look in to self-representation and maybe do it yourself.
In my experience the invoice would rarely go beyond the forwarder anyway. Usually just keep on file for the audit trail.
It sounds like the real problem is that the buyer doesn't want the consignee to know who their supplier (you) is but they want the goods delivering directly to them to save a bit of time and money. They are the middleman, buying and selling, who, like freight forwarders, often do not ever come in to physical contact with the goods.
Simple solution: request in writing that the freight forwarder only uses the invoice for domestic customs clearance and don't send any physical invoices with the goods.
Generally, but not always, this will be sufficient but sometimes overriding factors may come in to play. Example - Switzerland customs currently requires original invoices in order to customs clear goods for import. In reality, there is nothing to physically stopping the driver from handing over a copy on delivery - but if they did when you had asked them not to then the haulier can be pretty sure you won't be asking them to do the next run.
What I would say is that everyone involved within the supply chain is really working towards the same goals. By the very nature a certain amount of information has to be made available in order to make it possible for goods to move from one place to another. Without doing it all yourself, really all you can do is pick your suppliers well.